Rooted in a Bayesian framework, the TS algorithm [1]} defines an allocation policy by randomizing participants to arms in proportion to the posterior probability of each arm being superior at step \(t\) . In a two-armed setting, denoting by \(\pi _{t,1}^{\text{TS}}\)  TS's allocation probability of the experimental arm 1 at step \(t\) , and considering the reward specifications of Eq. (REF ), we have that:
\(\pi _{t,1}^{\text{TS}} & = \mathbb {P}\Big (\mathbb {E}\big (Y_{t,i}(A_{t,i} = 1)\big ) \ge \mathbb {E}\big (Y_{t,i}(A_{t,i} = 0)\big ) | \mathcal {H}_{t-1} \Big ) \nonumber \\ &= \mathbb {P}\Big (\mu _1 \ge \mu _0 | \mathcal {H}_{t-1} \Big ), \quad \quad \quad \quad \forall t \in [0,T]. \) 
